A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families
TL;DR: In this article, a new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used twoparameter families of life distributions as the Weibull, gamma and lognormal distributions.
Abstract: SUMMARY A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.
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01 Jan 2011
TL;DR: In this paper, a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions is presented.
Abstract: This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].
1,296 citations
TL;DR: A new two-parameter ageing distribution which is a generalization of the Weibull and able to model various ageing classes of life distributions including IFR, IFRA and modified bathtub (MBT), which provides an alternative to many existing life distributions.
251 citations
TL;DR: A short summary of some well-known, recent generations of Weibull-related lifetime models for quick information is presented and a clarification to a claim by Nadarajah & Kotz is offered.
Abstract: This short communication first offers a clarification to a claim by Nadarajah & Kotz. We then present a short summary (by no means exhaustive) of some well-known, recent generations of Weibull-related lifetime models for quick information. A brief discussion on the properties of this general class is also given. Some future research directions on this topic are also discussed.
219 citations
Cites result from "A new method for adding a parameter..."
...2 This claim is therefore inaccurate in view of our discussion above....
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TL;DR: In this article, a new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility, namely one-parameter exponential distribution, for data analysis purposes.
Abstract: A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.
211 citations
Cites background or methods from "A new method for adding a parameter..."
...Although Mudholkar and Srivastava (1993) proposed the exponentiated Weibull distribution, later several other exponentiated distributions have been introduced by several authors, see for example Gupta et al. (1998)....
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...Mudholkar and Srivastava (1993) proposed a method to introduce an extra parameter to a two-parameter Weibull distribution....
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...Mudholkar and Srivastava (1993)’s proposed exponentiated Weibull model has two shape parameters and one scale parameter....
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TL;DR: In this article, a new parametric distribution generated by Marshall and Olkin (1997) extended family of distributions based on the Lomax model was investigated and the proposed distribution can be expressed as a compound distribution with mixing exponential model.
Abstract: This paper investigates properties of a new parametric distribution generated by Marshall and Olkin (1997) extended family of distributions based on the Lomax model. We show that the proposed distribution can be expressed as a compound distribution with mixing exponential model. Simple sufficient conditions for the shape behavior of the density and hazard rate functions are given. The limiting distributions of the sample extremes are shown to be of the exponential and Frechet type. Finally, utilizing maximum likelihood estimation, the proposed distribution is fitted to randomly censored data.
203 citations
Cites methods from "A new method for adding a parameter..."
...1 provides another way to interpret the MOEL distribution in addition to the geometric-extreme stability property as given by Marshall and Olkin (1997)....
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References
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11,456 citations
TL;DR: In this article, the applicability of statistics to a wide field of problems is discussed, and examples of simple and complex distributions are given, as well as a discussion of the application of statistics in a wide range of problems.
Abstract: This paper discusses the applicability of statistics to a wide field of problems. Examples of simple and complex distributions are given.
9,091 citations
Book•
01 Jan 1994
TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
7,270 citations
Book•
01 Jan 1984
TL;DR: In this article, the authors give a concise account of the analysis of survival data, focusing on new theory on the relationship between survival factors and identified explanatory variables and conclude with bibliographic notes and further results that can be used for student exercises.
Abstract: The objective of this book is to give a concise account of the analysis of survival data. The book is intended both for the applied statistician and for a wider statistical audience wanting an introduction to this field. Particular attention is paid to new theory on the relationship between survival factors and identified explanatory variables. Each chapter concludes with bibliographic notes and outline statements of further results that can be used for student exercises. (ANNOTATION)
6,299 citations